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I have a class covering Power Spectral Density but I have no idea why it matters. Could someone provide some examples of its use?

Thanks

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closed as too broad by Marcus Müller, lennon310, jojek Mar 20 '18 at 11:38

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    $\begingroup$ Have you considered asking your professor? $\endgroup$ – MBaz Mar 19 '18 at 19:46
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    $\begingroup$ I second what @MBaz said: There's plenty of examples that you could find using google, and quite some that we might be able to give you, but it wouldn't make any sense in the context of the class you have. Ask your professor – we can't help you more than asking google would, and I don't think that under that condition, writing an answer would be a wise investment of your or our time. $\endgroup$ – Marcus Müller Mar 19 '18 at 21:52
  • $\begingroup$ It's a math course, professor doesn't care about applications. $\endgroup$ – FourierFlux Mar 20 '18 at 0:21
  • $\begingroup$ dunno that is true. $\endgroup$ – robert bristow-johnson Mar 20 '18 at 0:48
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This may sound a little glib, but actually it is not.

You know those little bars on some stereos that show you how much volume there is at each frequency band?

You're going to learn the math behind those, which also has applicability in many other places.

Ced

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  • $\begingroup$ one of the great examples which shows everyday use :) $\endgroup$ – arpit jain Mar 20 '18 at 6:53
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there are a couple of prerequisites.

1. first, do you know how to define a finite "power signal" vs. a finite "energy signal"?

2. here's another that's more explicit, but it's about discrete time, not continuous time.

3. and here is the beginning of a definition for Power Spectrum.

so repeating:

let $x(t)$ be a finite power signal. then, instantaneous power is:

$$ p_x(t) = |x(t)|^2 $$

the mean power is:

$$\begin{align} P_x &= \lim_{T \to \infty} \quad \frac{1}{T} \int\limits_{-\frac{T}{2}}^{+\frac{T}{2}} p_x(t) \, dt \\ \\ &= \lim_{T \to \infty} \quad \frac{1}{T} \int\limits_{-\frac{T}{2}}^{+\frac{T}{2}} |x(t)|^2 \, dt \\ \end{align}$$

now in this answer is some discussion of the sampling theorem, but i only want to use the brickwall filter results here. suppose $x(t)$ is passed through a brickwall filter having impulse response

$$ y(t) = h(t) \circledast x(t) $$

where

$$ h(t) = 2B \, \operatorname{sinc}(2B t) $$

and the frequency response is

$$ H(f) = \operatorname{rect} \left( \tfrac{f}{2B} \right) $$

so

$$ Y(f) = \begin{cases} X(f) \qquad & |f| \le B \\ 0 & |f| > B \\ \end{cases} $$

for any bandwidth $B>0$

the relation between Power Spectral Density to mean power is:

$$\begin{align} P_y &= \int\limits_{-\infty}^{\infty} S_y(f) \, df \\ &= \int\limits_{-B}^{+B} S_x(f) \, df \\ \end{align} $$

for any $B>0$.

what that means is $P_y$ is the power of $x(t)$ only at frequencies below $B$ in magnitude because all energy at frequencies above $B$ have been removed by the hypothetical brickwall low pass filter. because integral limits can add:

$$ \int\limits_{a}^{b} + \int\limits_{b}^{c} = \int\limits_{a}^{c}$$

that means that

$$ \int\limits_{f_1}^{f_2} S_x(f) \, df $$

is the power in $x(t)$ between the frequencies $f_1$ and $f_2$ (assuming $f_1 \le f_2$). suppose these two frequencies are very close. then:

$$S_x(f_0) \, \Delta f \approx \int\limits_{f_0}^{f_0+\Delta f} S_x(f) \, df $$

that means that $S_x(f_0)$ is proportional to the power around frequency $f_0$ and also proportional the tiny width of the frequency segment, $\Delta f$, around frequency $f_0$. this makes $S_x(f_0)$ a power density. it is the amount of power (in whatever units of power you make for $x(t)$) per unit frequency in the vicinity of $f_0$.

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  • $\begingroup$ Hmm, Ok Thanks. My issue is the term "spectrum" of a random process and "spectrum" of a realization of a signal are used interchangeably but they are not the same thing. Unless I am missing something, the "spectrum" of a random process doesn't say anything about the spectrum of the realizations and can't be used to divide the frequency domain for example, for communications. Is this right? $\endgroup$ – FourierFlux Mar 20 '18 at 4:00

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